Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions

This paper is concerned with periodic and antiperiodic boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues of these two different boundary value problems is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. In addition, a representation of solutions of a nonhomogeneous linear equation with initial conditions is given.     

Eigenvalues of second-order linear equations with coupled boundary conditions on time scales

In this paper we consider coupled boundary value problems for second-order linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only extend the existing ones of coupled boundary value problems for second-order difference equations but also contain more complicated time scales.     

Spectral theory of left definite difference operators

This paper is concerned with the spectral theory for the second-order left definite difference boundary value problems. Existence of eigenvalues of boundary value problems is proved, numbers of their eigenvalues are calculated and fundamental spectral results are obtained.     

具有混合型边界条件的左定Sturm-Liouville问题特征值的下标计算(英文)

本文研究具有混合型边界条件的左定Sturm-Liouvile问题特征值的下标计算问题.首先给出具有分离型边界条件和混合型边界条件的左定Sturm-Liouville问题的特征值之间的不等式;然后利用这个结果给出一种计算混合型边界条件下左定Sturm-Liouville问题特征值下标的方法.     

Matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions

An equivalence between a class of regular self-adjoint fourth-order boundary value problems with coupled or mixed boundary conditions and a certain class of matrix problems is investigated. Such an equivalence was previously known only in the second-order case and fourth-order case with separated boundary conditions.     

Spectral theory of discrete linear Hamiltonian systems

This paper is concerned with spectral problems for a class of discrete linear Hamiltonian systems with self-adjoint boundary conditions, where the existence and uniqueness of solutions of initial value problems may not hold. A suitable admissible function space and a difference operator are constructed so that the operator is self-adjoint in the space. Then a series of spectral results are obtained: the reality of eigenvalues, the completeness of the orthogonal normalized eigenfunction system, Rayleigh's principle, the minimax theorem and the dual orthogonality. Especially, the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated.     

New canonical forms of self-adjoint boundary conditions for regular differential operators of order four

In this paper, we find new canonical forms of self-adjoint boundary conditions for regular differential operators of order two and four. In the second order case the new canonical form unifies the coupled and separated canonical forms which were known before. Our fourth order forms are similar to the new second order ones and also unify the coupled and separated forms. Canonical forms of self-adjoint boundary conditions are instrumental in the study of the dependence of eigenvalues on the boundary conditions and for their numerical computation. In the second order case this dependence is now well understood due to some surprisingly recent results given the long history and voluminous literature of Sturm-Liouville problems. And there is a robust code for their computation: SLEIGN2.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Computing the indices of Sturm–Liouville eigenvalues for coupled boundary conditions (the EIGENIND-SLP codes)

The main purpose of the EIGENIND-SLP codes is to compute the indices of known eigenvalues of self-adjoint Sturm–Liouville problems with coupled boundary conditions (BCs). The spectrum of the problems can be unbounded from both below and above. Using some recent theoretical results, the computation is converted to that of the indices of the same eigenvalues for appropriate separated BCs, and is then carried out in terms of the Prüfer angle. The algorithm so generated and its implementation are discussed, and numerous examples are presented to illustrate the theoretical results and various aspects of the implementation.     

左定Sturm-Liouville算子的特征值计算(英文)

研究了定义在有限区间[a,b]上的具有分离型和混合型边界条件的左定正则Sturm-Liouville算子的特征值问题.把具有混合型边界条件的左定正则Sturm-Liouville问题转化成二维的、具有分离型边界条件的右定正则Sturm-Liouville问题,给出了具有混合型边界条件的左定正则Sturm-Liouville算子的特征值的数值计算方法.     

A note on relative oscillation theory for symplectic difference systems with general boundary conditions

We develop an analog of classical oscillation theory for discrete symplectic eigenvalue problems with general self-adjoint boundary conditions which, rather than measuring of the spectrum of one single problem, measures the difference between the spectra of two different problems. We prove formulas connecting the numbers of eigenvalues in a given interval for two symplectic eigenvalue problems with different self-adjoint boundary conditions. We derive as corollaries generalized interlacing properties of eigenvalues.     

Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations

This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace (relation) below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspace, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.     

Stable difference schemes for certain parabolic equations

In some applications, boundary value problems for second-order parabolic equations with a special nonself-adjoint operator have to be solved approximately. The operator of such a problem is a weighted sum of self-adjoint elliptic operators. Unconditionally stable two-level schemes are constructed taking into account that the operator of the problem is not self-adjoint. The possibilities of using explicit-implicit approximations in time and introducing a new sought variable are discussed. Splitting schemes are constructed whose numerical implementation involves the solution of auxiliary problems with self-adjoint operators.     

High order approximations of the eigenvalues of sturm-liouville problems with coupled self-adjoint boundary conditions

Sampling theory has been used to compute with great accuracy the eigenvalues of regular and singular Sturm-Liouville problems of Bessel Type. We shall consider in this paper the case of general coupled real or complex self-adjoint boundary conditions. We shall present few examples to illustrate the power of the method and compare our results with the ones obtained using the well-known Sleign2 package.     

Primal-dual variational problems by boundary and finite elements

In this paper the primal-dual (or mixed) formulation is studied for self-adjoint elliptic problems coupled with a boundary integral equation. It is shown that, after introducing a suitable complementary variational principle, the problem is reduced to finding a stationarity point of a constrained functional. Some numerical examples are reported for a second-order differential equation on unbounded domains.     

Matrix representations of fourth order boundary value problems with finite spectrum

We show that a class of regular self-adjoint fourth order boundary value problems is equivalent to a certain class of matrix problems. Equivalent here means that they have exactly the same eigenvalues. Such an equivalence was previously known only in the second order case.     

An eigenvalue comparison theorem for second order self-adjoint differential equations

Summary This paper is concerned with a comparison of the eigenvalues for pairs of self-adjoint differential systems which arise from a single ordinary second-order differential equation. The sistems differ in the boundary conditions which are imposed, and it is these boundary conditions which will draw most of the attention. The principal results obtained deal with predicting alternation of the eigenvalues for two such systems from the boundary conditions alone, without special consideration of the differential equation. This paper is part of a thesis submitted to Carnegie Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author wishes to express his thanks to the thesis director, ProfessorAllan D. Martin Jr. Presented to the American Mathematical Society November 17, 1962.     

Spectral theory of Sturm–Liouville difference operators

We present several classes of explicit self-adjoint Sturm–Liouville difference operators with either a non-Hermitian leading coefficient function, or a non-Hermitian potential function, or a non-definite weight function, or a non-self-adjoint boundary condition. These examples are obtained using a general procedure for constructing difference operators realizing discrete Sturm–Liouville problems, and the minimum conditions for such difference operators to be self-adjoint with respect to a natural quadratic form. It is shown that a discrete Sturm–Liouville problem admits a difference operator realization if and only if it does not have all complex numbers as eigenvalues. Spectral properties of self-adjoint Sturm–Liouville difference operators are studied. In particular, several eigenvalue comparison results are proved.     

Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators

For boundary value problems generated by a second-order differential equation with regular nonseparated boundary conditions, criteria for the eigenvalues to be multiple are given and the relative position of the eigenvalues is studied. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 369–381, March, 2000.     

Stability and instability regions of second-order equations with periodic point interactions on time scales

This paper studies the stability and instability regions of second-order equations with periodic point interaction on time scales. By the Floquet theory and the inequalities among eigenvalues of second-order equations with coupled boundary conditions, our main results are obtained.